Diophantine sets are subsets defined by polynomial equations over integers. More precisely We obtain them basically by projecting integers solutions of some polynomial equations. Observation 1: Using the statement iff we see that there is no difference between one polynomial equation and a system of polynomial equations. Example: Even numbers are Diophantine: Odd numbers: Non-negative […]
Category: Number Theory
Mordell’s Equation
The equation is called Mordell’s equation. For , it is non-singular and defines an elliptic curve (genus 1, group structure). Rational Points: We first describe the torsion part of the rational solutions. By Nagell-Lutz, these solutions should have integer coordinates. Torsion: Assume that is free of sixth powers. If not we can use the change […]
Sophie Germain’s Theorem
As early attempts to the proof of Fermat’s last theorem, many mathematicians solved the problem for small exponents. While these special cases are being studied, Sophie Germain, a French mathematician, came up with the following interesting result. (Look at https://www.agnesscott.edu/lriddle/women/germain.htm for her fascinating and revolutionary story) Theorem 1: For any odd prime such that is […]
Kürschâk and Nagel’s theorems (Erdos 1932)
None of the above quantities are integers. Proof: For the first expression, look at the largest prime- when we clear denominators, the denominator is divisible by this prime and numerator is not.For the second expression, if the is smaller then then the quantity is less than one, otherwise there will be a prime between $m$ […]
Betrand Postulate- Tchebyshev estimates – Erdos( 1932 -1)
Basic idea to approximately count primes is to look at and look at it prime factorization. To get access to primes between and , we need to look at the binomial coefficients Another way to see it: Use and sum over . Use the resulting expression to estimate the number of primes. (This is just […]
Zeta(2)
Basel Problem 1644: Find the value of Euler (1735) showed that Euler’s Proof: But Comparing the coefficients of we get, Fourier Proof: Consider on as a periodic function. Parseval gives Apostol’s Proof: To compute the integral using the change of coordinates to get Using we get The integral are computed below. Hence
Eisenstein Lattice Proof of Quadratic Reciprocity:
Let be distinct odd primes. Start with the following expression for Legendre symbol. which detects if is square modulo or not. By writing , and multiplying the quantities , it is easy to see that where the sum is over even integers . We used that is just a permutation of Therefore we get Now […]
A Neighbourhood Of Infinity 